Infi-1 13

\begin{matrix} lim_{n \to \infty} a_{n} = L ⟺ \forall K < L, W > L : \exists N_{ε} : \forall n > N_{ε} : K < a_{n} < W \\ Let ε = m i n (\frac{L - K}{2}, \frac{W - L}{2}) \\ \exists N_{ε} : \forall n > N_{ε} : K < L - ε < a_{n} < L + ε < W \end{matrix}

Proof for limit comparison test

\begin{matrix} 0 \leq a_{n}, b_{n} \\ lim_{n \to \infty} \frac{a_{n}}{b_{n}} = L \end{matrix}

\begin{matrix} Proof for 1. \\ L > 0 \\ ⟹ \frac{1}{e} L < \frac{a_{n}}{b_{n}} < e L \\ \frac{1}{e} L b_{n} \leq a_{n} \leq e L b_{n} \\ 1. \sum a_{n} \to M_{1} ⟹ \sum \frac{1}{e} b_{n} \to M^{'} ⟹ \sum b_{n} \to M_{2} \\ 2. \sum b_{n} \to M_{2} ⟹ \sum e L b_{n} \to M^{'} ⟹ \sum a_{n} \to M_{1} \\ 1. and 2. ⟹ \sum a_{n} \to M_{1} ⟺ \sum b_{n} \to M_{2} \end{matrix}

\begin{matrix} Proof for 2. \\ L = 0, \sum b_{n} \to M_{2} \\ \frac{a_{n}}{b_{n}} \to 0 ⟹ \frac{a_{n}}{b_{n}} < 7 ⟹ a_{n} < 7 b_{n} \\ \sum b_{n} \to M_{2} ⟹ \sum 7 b_{n} \to 7 M_{2} ⟹ \sum a_{n} \to M_{1} \end{matrix}

\begin{matrix} Proof for 3. \\ L = \infty, \sum a_{n} \to M_{1} \\ \frac{a_{n}}{b_{n}} \to \infty ⟹ \frac{a_{n}}{b_{n}} > 420 ⟹ a_{n} > 420 b_{n} \\ \sum a_{n} \to M_{1} ⟹ \sum 420 b_{n} \to M^{'} ⟹ \sum b_{n} \to M_{2} \end{matrix}

\begin{matrix} Let L = lim_{n \to \infty} \sqrt[n]{| a_{n} |} \\ L > 1 ⟹ \sum a_{n} ↛ M \\ L < 1 ⟹ \sum | a_{n} | \to M \\ L = 1 ⟹ Test is inconclusive \\ Proof: \\ Let \overset{―}{lim_{n \to \infty}} \sqrt[n]{| a_{n} |} \\ ⟹ \exists a_{k_{n}} : \sqrt[k_{n}]{| a_{k_{n}} |} \to L \\ If L > 1 ⟹ L > \frac{L + 1}{2} > 1 \\ ⟹ \sqrt[k_{n}]{| a_{k_{n}} |} \geq \frac{L + 1}{2} ⟺ | a_{k_{n}} | \geq {(\frac{L + 1}{2})}^{k_{n}} \\ \frac{L + 1}{2} > 1 ⟹ {(\frac{L + 1}{2})}^{k_{n}} \to \infty ⟹ | a_{k_{n}} | \to \infty \\ ⟹ a_{k_{n}} ↛ 0 ⟹ a_{n} ↛ 0 ⟹ \sum a_{n} ↛ M \\ If L < 1 ⟹ L < \frac{L + 1}{2} \\ \sqrt[k_{n}]{| a_{k_{n}} |} \to L ⟹ \sqrt[n]{| a_{n} |} \leq \frac{L + 1}{2} ⟹ | a_{n} | \leq {(\frac{L + 1}{2})}^{n} \\ \sum {(\frac{L + 1}{2})}^{n} \to M^{'} ⟹ \sum | a_{n} | \to M \end{matrix}

\begin{matrix} Let L = lim_{n \to \infty} | \frac{a_{n + 1}}{a_{n}} | \\ L > 1 ⟹ \sum a_{n} ↛ M \\ L < 1 ⟹ \sum | a_{n} | \to M \end{matrix}